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OPEN Quasiparticle Properties under Interactions in Weyl and Nodal Line Semimetals Received: 23 July 2018 Jing Kang 1, Jianfei Zou1, Kai Li2, Shun-Li Yu3,4 & Lu-Bing Shao3,4 Accepted: 22 January 2019 The quasiparticle spectra of interacting Weyl and nodal-line semimetals on a cubic lattice are studied Published: xx xx xxxx using the cluster perturbation theory. By tracking the spectral functions under interaction, we fnd that the Weyl points will move to and meet at a specifc point in one Weyl semimetal model, while in the other Weyl semimetal model they are immobile. In the nodal-line semimetals, we fnd that the nodal line shrinks to a point and then disappears under interaction in one-nodal-line system. When we add another nodal line to this system, we fnd that the two nodal lines both shrink to specifc points, but the disappearing processes of the two nodal lines are not synchronized. We argue that the nontrivial evolution of Weyl points and nodal lines under interaction is due to the presence of symmetry breaking order, e.g., a ferromagnetic moment, in the framework of mean feld theory, whereas the stability of Weyl points under interaction is protected by symmetry. Among all these models, the spectral gap is fnally opened when the interaction is strong enough.

Te discovery of semimetallic features in electronic band structures protected by the interplay of symmetry and topology provides a realistic platform for the concepts of fundamental physics theory in condensed matter phys- ics. In recent years, Weyl semimetals1–4 (WSM) have been attracted considerable attention since they extend the topological classifcation of mater beyond the insulators and exhibit the exotic Fermi arc surface state5. WSM 6,7 8,9 materials such as the TaAs-family pnictides and MoTe2 have been discovered by observation of the unique Fermi arcs of surface states through angle-resolved photoemission spectroscopy10–15. In these materials, the con- duction and valence bands touch each other around several points, which are called Weyl points (WPs), in the momentum space. Te WP acts as a topological monopole which can be quantifed by corresponding chiral charge through calculating the fux of Berry curvature16. According to the conservation of chirality, the WPs always exist in pairs of opposite chirality. Te topological Fermi arcs are arising from the connection of two projections of the bulk WPs with two opposite chiral charges in the surface Brillouin zone (BZ). WSMs and their surface states may lead to unusual spectroscopic and transport phenomena such as chiral anomaly, spin and anomalous Hall efects9. Te WSMs have a Fermi surface consisting of a fnite number of WPs in the BZ, at which the conduction and valence bands meet linearly. Te WPs are twofold degeneracy and only exist in condensed matter systems with breaking either time-reversal or spatial-inversion symmetry. Such a phase has massless Weyl quasiparticles which can be viewed as half-Dirac Fermions. Besides, the Weyl quasiparticles can be gapped by coupling two quasiparticles with diferent chirality17. In the presence of interactions which can easily arise in realistic systems, the Weyl quasiparticles can be moved, normalized and even gapped. Another interesting system is the nodal-line semimetal(NLSM) with one-dimensional Fermi surfaces18–21. In the experiment side, the NLSMs were observed 22 23,24 in several compounds such as PbTaSe2 and ZrSiS . In this system there are bulk band touchings along 1D lines and these line-like touchings need extra symmetries to be topologically protected. Interactions can also be applied in this system to discuss the proximity efect and spontaneous symmetry breaking25. In this paper, by using the Cluster Perturbation Teory (CPT)26–31, we study two lattice models for WSMs and one for NLSM to see how the on-site Coulomb interaction afects the WPs and nodal lines. In CPT, the quasipar- ticle spectral function can be calculated and then the positions of the WPs and nodal lines can be tracked through

1College of Science, Hohai University, Nanjing, 210098, China. 2School of Physics and Engineering, Zhengzhou University, Zhengzhou, 450001, China. 3National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, 210093, China. 4Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China. Correspondence and requests for materials should be addressed to J.K. (email: [email protected]) or K.L. (email: [email protected])

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Figure 1. (a) A 2 × 2 × 2-site cluster, as marked by numbers 1–8, in the cubic lattice. (b) Te scanning routine is Γ-X-M-Γ-R-X′-Γ in the bulk BZ.

the spectral function when the interaction alters. We fnd that the WPs will move to and meet each other under interactions in one WSM model, while in the other WSM model they will not. In the one-nodal-line semimetal system, the nodal line shrinks to a point under interactions. When we add another nodal line to this system, we fnd that the two nodal lines will all shrink to specifc points one afer the other. We argue that the nontrivial evolution of WPs and nodal lines under interaction is due to the presence of symmetry breaking order, e.g., a ferromagnetic moment, in the framework of mean feld theory, whereas the stability of WPs under interaction is protected by symmetry. Among all these models, the spectral gap is fnally opened when the interaction is strong enough. Models and Methods Weyl Semimetal Models. We will consider two kinds of Weyl Semimetal Models. For the frst kind of WSM model (WSM1), the tight-binding Hamiltonian is written as follows16,32:

=−† +− − σ Hc00∑k kx{[2(tkcoscoskm)(2coskkyzcos)] x ++2stkin yyσσ2stkin zz},ck (1)

††† where ckk= (,cc↑↓k ) and σx,y,z are the Pauli matrices. Te hopping constant t will be set t = 1 in calculations based on this model. Tis model breaks both time-reversal and space-inversion symmetries. Afer diagonalizing this non-interacting Hamiltonian, we can get the band structure

=± −+−− 22++2 . εkx[2(coskkcos)0 mk(2 coscyzoskk)] 4sin yz4sin k (2)

Assuming εk = 0, we get the WPs in the bulk BZ. Tere can be 2, 6 or 8 WPs for diferent parameter m, while two of the WPs are present at (±k0, 0, 0). We now introduce the second kind of WSM model (WSM2) which, in contrast to both the WSM1 and the second Weyl semimetal model studied in ref.16, preserves the space-inversion symmetry but not the time-reversal symmetry. Te tight-binding Hamiltonian is written as follows:

=+† σσ+.σ Ht0 2[∑kckxcos(kkk)cxyos()yzcos( )]zkc (3) Similar to the second Weyl semimetal model studied in ref.16, the real space hopping is a type of bond-selective and spin-dependent Kitaev-like hopping31. Afer diagonalizing this non-interacting Hamiltonian, we can get the band structure

=± 222++. εkx2ct os kkkcoscyzos (4) πππ Tis model gives us eight WPs at ±±,,± . ( 222) Nodal-Line Semimetal Model. For the NLSM model, the tight-binding Hamiltonian is written as follows25:

† Hc01=+∑ kx{2[(tkcoscoskbyz−+)(tk23cos1−+)]σσxz2stkin yk},c k (5)

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Figure 2. Intensity map of the spectral function A(k, ω) along high symmetry lines (see Fig. 1(b)) with m = 1.5 and k0 = 0.2π based on the WSM1 model for (a) U = 0, (b) U = 3, (c) U = 8 and (d) U = 9, respectively. Te red dashed lines denote the non-interacting band structures.

where t1, t2, t3 are the hopping constants. We can also diagonalize the non-interacting Hamiltonian to get the band structure, which reads:

=± +−+−2 +.22 εkx2[tk12(cos cos)kbyztk(cos 1)]stk3 in z (6)

During our calculations based on this model, the hopping constant t1 will be set as the energy unit t1 = 1. By setting proper parameters t2, b, we can get one or two nodal lines. Interaction Hamiltonian. In order to investigate the efects of interactions in the semimetals, we use the Hubbard model =+ − μ HH0,Un∑∑i ii↑↓nn,,i i σ, (7)

where H0 denotes a tight-binding Hamiltonian of the non-interacting semimetals as we have introduced above. U is the on-site Coulomb interaction, ni,σ is the particle-number operator on site i with spin σ and μ is the chemical potential which will be fxed μ = 1 U at half-flling as the band structures hold particle-hole symmetry. 2 Numerical Method. A good way to analyze the Hubbard model numerically is the exact diagonalization (ED). However, due to the limitation of computer memory capacity and speed of CPU, the solvable size of the lat- tice cannot be large. Tus one will not get enough momentum points to study the k-space distribution of the spec- tral function. Te CPT which is based on the ED is another numerical method to solve the Hubbard model26–28. Te Green’s function in a fnite size of lattice is frst calculated through the ED. Ten we can divide the whole lattice into many small lattices of this size which are usually called clusters. Te inter-cluster hopping is treated perturbatively and the Green’s function of the physical system is obtained from strong-coupling perturbation theory. As we get enough momentum points, we can track the evolution of WPs or nodal lines controlled by the variation of U. During our calculations, we choose a 2 × 2 × 2-site cluster in the cubic lattice shown in Fig. 1(a). From the exact diagonalization, the Green’s function in each cluster can be calculated from its defnition

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Figure 3. Intensity map of the spectral function A(k, ω) along high symmetry lines based on the WSM1 model: (a–c) k0 = 0.5π and U = 0, 3, 5, respectively; (d–f) k0 = 0.9π and U = 0, 0.4, 1, respectively. Te red dashed lines represent the non-interacting band structures.

1 † Gc′=μσ,νσ′′()ω 〈Ω| μσ cνσ |Ω〉 ωη−+HE0 + i † 1 +〈Ω|cνσ′ cμσ|Ω〉, ωη+−HE0 + i

where μ and ν denote diferent lattice sites within a cluster with spin index σ, σ′ and E0 the energy of the ground state. Te system Green function G is obtained from the cluster Green function G′ using the random phase like approximation

Gk(,ωω)(=−GV′′)[1(kG)(ω)]−1,(8)   0Rki⋅R where k belongs to the reduced Brillouin zone corresponding to the superlattice and VVμσ,,νσ′′()k = ∑R μσ νσ e 0R with R the superlattice index. Vμσ,νσ is the hopping constant from sub-lattice site μ with spin σ to site ν with spin σ′ between the original superlattice 0 and superlattice R. Te k-dependent Green function Gcpt(k, ω) is given by

cpt 1 −⋅ikr()μν−r  Gσσ′(,kkωω) = ∑eGμσ,νσ′(, ), L μν (9) where kkK=+ with K the reciprocal vector of the superlattice. Afer getting the system Green’s function by CPT, we study the spectral function of quasiparticles which can be obtained by

cpt AG(,kkωω)I=−∑ m(σσ ,)/π. σ (10) We have checked the results by varying the cluster size and found no qualitative diference.

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Figure 4. Intensity map of the spectral function A(k, ω) along high symmetry lines based on the WSM2 model for: (a) U = 0; (b) U = 2; (c) U = 8, respectively. (d) Te imaginary part of the self-energy at the Weyl points for WSM1 (k0 = 0.9π) and WSM2 models when ω = 0.

Results and Discussion As for the WSM1 model, where the number of WPs is determined by the parameter m, we frst set m = 1.5 so that the system has only two WPs which exist at (±k, 0, 0). Intensity plots of A(k, ω) along high symmetry lines are shown in Fig. 2, which gives a clearly evolution of the WPs for diferent interactions, as the WPs are initialized at (±0.2π, 0, 0). For the non-interacting state (U = 0, Fig. 2(a)), the lower and upper band touch at the WP. When we increase interaction strength U, the WP shifs along the kx direction towards X point and no gap exists in this state. Finally the WPs meet at X point when U reaches 8 with no band gap. As U continues growing, a gap will be opened at X point, e.g., U = 9 shown in Fig. 2(d). Te magnitude of the gap will also rise with U and the system becomes an insulator. We then change the value of k0 to see whether this phenomena is a special case. Te results are shown in Fig. 3 for k0 = 0.5π (a–c) and k0 = 0.9π (d–f). We can see that the WPs are pushed to the X point and meet together by increasing U. Afer that a gap comes out and its magnitude increases with U. Te diference among the results is that the nearer X point the WP is, the smaller U the gap opens for. Another parameter we can change in this model is m, which afects the bandwidth and number of the WPs. If we increase m, the number of WPs will not change and the bandwidth will become larger. Te movement of WPs is just the same under the present of U, as no qualitative change can be observed in contrast with m = 1.5. If m is decreased, the number of WPs will change to 6 or 8. Te movement of WPs is similar under the present of U. One possible explanation of the above movement of WPs is due to the mean-field analysis in the weak-interaction regime, as discussed in refs16,33. Specifcally, the Hubbard interaction can be decoupled as

U † 2 Unii,,↑↓n →+()nnii,,↑↓−+Umxiccσxi Umx , 2 (11) where m =〈1 cc†σ 〉 represents the magnetization order parameter along the spin-x direction. For Eq. 1, there is xi2 xi † already a ferromagnetic moment 2(cmt− oskc0)∑i ixσ ci at U = 0, where 2(cmt−>osk0)0 within our calcula- tions for the two WPs case. At the mean-feld level, the magnetization is enhanced by turning on the repulsive U, i.e.,

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Figure 5. Intensity map of the spectral function A(k, ω) along high symmetry lines based on the NLSM model. Evolution of A(k, ω) in the one-nodal-line state is shown in (a–c) with parameters t2 = 1, b = 1.5 and (a) U = 0; (b) U = 1; (c) U = 3, respectively. (d–i) show the evolution of A(k, ω) in the two-nodal-line state with parameters t2 = 0.5, b = 0.3 for (d) U = 0; (e) U = 1; (f) U = 2; (g) U = 3; (h) U = 5; (i) U = 8, respectively. Te insets show the Fermi surfaces in kz = π or kz = 0 planes.

−Umx > 0 increases with U. Terefore, we see that the two WPs move to larger k0 (e.g., cosckk00→+os Umx) with increasing U. We now turn to the WSM2 model. In this model, there are eight WPs. When the interaction U is introduced, the positions of WPs do not change and a gap is opened at the WP when U is large enough, as shown in Fig. 4. A possible explanation is again due to the mean-feld argument: Since there is no magnetization at U = 0, turning on a weak repulsive U would not induce a magnetization at the mean-feld level, e.g., Umx = 0, and hence the posi- tions of the WPs are not afected. On the other hand, the stability of WPs under the Hubbard interaction (below a critical U) may be protected by the space-inversion symmetry of this model. Now, a few remarks are in order concerning the nature of the metal-insulator transition in our WSM models33. Te gap in WSM1 model opens afer the WPs merge at the X point, especially when the non-interacting WPs are very close to each other, a small U could open the gap. While in the WSM2 model, the gap directly opens at the WPs until U is large enough (see Fig. 4). For large U the insulating phase should be a Mott insulator, which can be identifed by checking whether the imaginary part of the self-energy at low frequency diverges33. Te numerical results of the imaginary part are shown in Fig. 4(d). We see that, for the WSM2 model, the imaginary part grows very fast afer the opening of the gap at U = 4 (we do not see a sudden divergence due to the fnite-size efect). Tis divergence indicates that the gap should be a Mott gap for the WSM2 model. Whereas for the WSM1 model, the imaginary part of the self-energy has no divergence when the gap opens, and this behavior persists within a large range of U. Obviously, this is not a Mott gap. When U reaches about 4, it starts to diverge rapidly, which means that the system enters the Mott insulating phase. In the following, we will discuss the NLSM model. Tis model contains three adjustable parameters t2, t3 and b. One of them, t3, will not afect the positions of nodal line and will be set t3 = 1 during our calculations. b controls

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the shape of nodal line and t2 determines the number of nodal lines. Afer choosing suitable parameters, we get one-nodal-line and two-nodal-line states. We frst set the parameters t2 = 1 and b = 1.5 under which the system enters the one-nodal-line state. We track the evolution of A(k, ω) by changing the interaction strength U. For the non-interaction state, we can see a nodal line in the kz = 0 plane and the upper and lower bands touch each other along the line (shown in Fig. 5(a)). As U increases, e.g., U = 1, we can see clearly in Fig. 5(b) that the position of the nodal line changes and the nodal line starts to shrink to the Γ point. Tere is no band gap and the system remains as a semimetal. With a further increase of U, the nodal line continues its shrinking and fnally become a point at (0, 0, 0). Afer that, if we keep increasing U, a gap will be opened at the Γ point and the system become an insulator (see Fig. 5(c)). Te evolution of A(k, ω) shows very similar as what we have discussed in the WSM1 model with two WPs. Next we will discuss the results for the existence of two nodal lines. The parameters are set t2 = 0.5 and b = 0.3. Tere are two nodal lines: one lies in the kz = 0 plane, while the other lies in the kz = π plane. For the non-interaction state, shown in Fig. 5(d), we can see the nodal lines clearly. When an interaction U is introduced in this system, such as U = 1, 2 and 3, one nodal line lying in the kz = π plane begins to shrink toward X′ point, while the other lying in the kz = π plane shrinks toward Γ point (see Fig. 5(e–g)). As U continues growing, the nodal line in the kz = π plane becomes a point at X′ point, and then a gap opens at the this point. However, the system remains as a semimetal, as the nodal line in the kz = 0 plane still exists. When U is further increased, the nodal line in the kz = 0 plane continues its shrinking and the magnitude of the gap at X′ point increases, as shown in Fig. 5(h) with U = 5 for example. When U reaches around 8, both of the nodal lines disappear and a full gap is opened. Te system fnally goes into the insulating state. Te positions of nodal lines in the momentum space can be determined by equations cos kx + cos ky = b for kz = 0 plane and cos kx + cos ky = b + 2t2 for kz = π plane. When U is introduced into this system, the parameters t2 and b will be renormalized to larger magnitudes. As a result, the nodal lines will shrink to X′ point in the kz = π plane and Γ point in the kz = 0 plane. Moreover, due to the extra parameter 2t2, the nodal line in the kz = π plane shrinks faster than that in the kz = 0 plane. To be more precise, let us elaborate this using the mean-feld analysis, † e.g., Eq. 11. As can be seen from Eq. 5, there is already a ferromagnetic moment ()−−tb12tc∑i ixσ ci at U = 0, where (−t1b − t2) < 0 within our calculations for the NLSM model. At the mean-feld level, the magnitude of magnetization is enhanced by turning on the repulsive U (i.e., Umx > 0 increases with U), which efectively enlarge the parameters t2 and b as: b →+bUmbxx,2+→tb22++2tUm . Finally, we would like to discuss the efect of quasi-particle weight Z which is ignored in our mean-feld analysis33. In the presence of the Hubbard interaction, the single-particle band structure is renormalized by the factor Z. Within the framework of mean-feld theory, the low-energy efective Hamiltonian can be approximated as Hef = ZHMF, where HMF denotes the mean-feld Hamiltonian obtained from Eq. 11. Tus, we see that the quasi-particle weight Z does not afect the positions of Weyl nodes and the form of nodal lines in our models. Summary In summary, we have studied the evolutions of the WPs and nodal lines under interaction U using CPT. For WSM1 model, the WPs move towards a specifc point with the increase of U. When the WPs meet at the point, a gap is opened there and the system becomes an insulator. In WSM2 model, the WPs are static and when U is strong enough, a full gap is opened at the WPs. In the NLSM model, we have discussed the one-nodal-line and two-nodal-line states. For the one-nodal-line state, the only nodal line shrinks to a specifc point in its plane and fnally a gap is open at the point. For the two-nodal-line state, the two nodal lines both shrink but not in step. When both nodal lines disappear, a full gap is opened with increasing U. We argue that the nontrivial evolution of WPs and nodal lines under interaction is due to the presence of symmetry breaking order, e.g., a ferromagnetic moment, in the framework of mean feld theory, whereas the stability of WPs under interaction is protected by symmetry. Among all these models, the spectral gap is fnally opened when the interaction is strong enough. References 1. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011). 2. Balents, L. Viewpoint: Weyl electrons kiss. Physics 4, 36 (2011). 3. Yan, B. & Fesler, C. Topological Materials: Weyl Semimetals. Annu. Rev. Condens. Matter Phys. 8, 337 (2017). 4. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018). 5. Togo, A., Chaput, L., Tanaka, I. & Hug, G. 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